# American Institute of Mathematical Sciences

December  2021, 3(4): 677-700. doi: 10.3934/fods.2021017

## Homotopy continuation for the spectra of persistent Laplacians

 1 Department of Mathematics, Michigan State University, MI 48824, USA 2 Department of Mathematics, Department of Electrical and Computer Engineering, Department of Biochemistry and Molecular Biology, Michigan State University, MI 48824, USA

* Corresponding author: Guo-Wei Wei

Received  March 2021 Revised  June 2021 Published  December 2021 Early access  July 2021

Fund Project: This work was supported in part by NIH grant GM126189, NSF grants DMS-2052983, DMS-1761320, and IIS-1900473, NASA grant 80NSSC21M0023, Michigan Economic Development Corporation, George Mason University award PD45722, Bristol-Myers Squibb 65109, and Pfizer. The authors thank Dr. Wenrui Hao and Ms. Rui Wang for discussion and/or help

The $p$-persistent $q$-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the $q$-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.

Citation: Xiaoqi Wei, Guo-Wei Wei. Homotopy continuation for the spectra of persistent Laplacians. Foundations of Data Science, 2021, 3 (4) : 677-700. doi: 10.3934/fods.2021017
##### References:
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Love, B. Filippenko, V. Maroulas and G. Carlsson, Topological deep learning, preprint, arXiv: 2101.05778. Google Scholar [24] F. Mémoli, Z. Wan and Y. Wang, Persistent Laplacians: Properties, algorithms and implications, preprint, arXiv: 2012.02808. Google Scholar [25] Z. Meng, D. Vijay Anand, Y. Lu, J. Wu and K. Xia, Weighted persistent homology for biomolecular data analysis, Scientific Reports, 10 (2020), 1-15. doi: 10.1038/s41598-019-55660-3.  Google Scholar [26] F. Nasrin, C. Oballe, D. Boothe and V. Maroulas, Bayesian topological learning for brain state classification, 18th IEEE International Conference On Machine Learning And Applications (ICMLA), Boca Raton, FL, 2019. doi: 10.1109/ICMLA.2019.00205.  Google Scholar [27] D. D. Nguyen, Z. Cang and G.-W. Wei, A review of mathematical representations of biomolecular data, Phys. Chem. Chem. Phys., 22 (2020), 4343-4367.  doi: 10.1039/C9CP06554G.  Google Scholar [28] Y. Ren, J. W. R. Martini and J. Torres, Decoupled molecules with binding polynomials of bidegree $(n, 2)$, J. Math. Biol., 78 (2019), 879-898.  doi: 10.1007/s00285-018-1295-x.  Google Scholar [29] I. Sgouralis, A. Nebenführ and V. Maroulas, A Bayesian topological framework for the identification and reconstruction of subcellular motion, SIAM J. Imaging Sci., 10 (2017), 871-899.  doi: 10.1137/16M1095755.  Google Scholar [30] A. J. Sommese and C. W. Wampler II, The Numerical Solution of Systems of Polynomials. Arising in Engineering and Science, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/5763.  Google Scholar [31] J. Townsend, C. P. Micucci, J. H. Hymel, V. Maroulas and K. D. Vogiatzis, Representation of molecular structures with persistent homology for machine learning applications in chemistry, Nature Communications, 11 (2020), 1-9.  doi: 10.1038/s41467-020-17035-5.  Google Scholar [32] J. Verschelde, Algorithm 795: Phcpack: A general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Softw., 25 (1999), 251-276.  doi: 10.1145/317275.317286.  Google Scholar [33] C. W. Wampler and A. J. Sommese, Numerical algebraic geometry and algebraic kinematics, Acta Numer., 20 (2011), 469-567.  doi: 10.1017/S0962492911000067.  Google Scholar [34] R. Wang, D. D. Nguyen and G.-W. Wei, Persistent spectral graph, Int. J. Numer. Methods Biomed. Eng., 36 (2020), 27pp. doi: 10.1002/cnm.3376.  Google Scholar [35] R. Wang, R. Zhao, E. Ribando-Gros, J. Chen, Y. Tong and G.-W. Wei, HERMES: Persistent spectral graph software, Foundations of Data Science, 3 (2020), 67-97.  doi: 10.3934/fods.2021006.  Google Scholar [36] K. Xia and G.-W. Wei, Persistent homology analysis of protein structure, flexibility, and folding, Int. J. Numer. Methods Biomed. Eng., 30 (2014), 814-844.  doi: 10.1002/cnm.2655.  Google Scholar [37] X.-D. Zhang, The Laplacian eigenvalues of graphs: A survey, preprint, arXiv: 1111.2897. Google Scholar

show all references

##### References:
 [1] E. L. Allgower, D. J. Bates, A. J. Sommese and C. W. Wampler, Solution of polynomial systems derived from differential equations, Computing, 76 (2006), 1-10.  doi: 10.1007/s00607-005-0132-4.  Google Scholar [2] D. N. Arnold, G. David, M. Filoche, D. Jerison and S. Mayboroda, Computing spectra without solving eigenvalue problems, SIAM J. Sci. Comput., 41 (2019), B69–B92. doi: 10.1137/17M1156721.  Google Scholar [3] D. J. Bates, I. A. Fotiou and P. Rostalski, A numerical algebraic geometry approach to nonlinear constrained optimal control, 46th IEEE Conference on Decision and Control, New Orleans, LA, 2007. doi: 10.1109/CDC.2007.4434470.  Google Scholar [4] D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Bertini: Software for numerical algebraic geometry., Available from: https://bertini.nd.edu. Google Scholar [5] D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Numerically Solving Polynomial Systems with Bertini, Software, Environments, and Tools, 25, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. doi: 10.1137/1.9781611972702.  Google Scholar [6] P. Breiding and S. Timme, HomotopyContinuation.jl: A package for homotopy continuation in Julia, in International Congress on Mathematical Software, Lecture Notes in Computer Science, 10931, Springer, 2018,458–465. doi: 10.1007/978-3-319-96418-8_54.  Google Scholar [7] Z. Cang, L. Mu, K. Wu, K. Opron, K. Xia and G.-W. Wei, A topological approach for protein classification, Computational and Mathematical Biophysics, 3 (2015), 140-162.  doi: 10.1515/mlbmb-2015-0009.  Google Scholar [8] G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308.  doi: 10.1090/S0273-0979-09-01249-X.  Google Scholar [9] T. Chen, T.-L. Lee and T.-Y. Li, Hom4ps-3: A parallel numerical solver for systems of polynomial equations based on polyhedral homotopy continuation methods, in Mathematical Software – ICMS 2014, Lecture Notes in Comput. Sci., 8592, Springer, Heidelberg, 2014,183–190. doi: 10.1007/978-3-662-44199-2_30.  Google Scholar [10] H. Edelsbrunner and J. L. Harer, Computational Topology: An Introduction, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/mbk/069.  Google Scholar [11] J. Friedman, Computing betti numbers via combinatorial Laplacians, Algorithmica, 21 (1998), 331-346.  doi: 10.1007/PL00009218.  Google Scholar [12] M. Gameiro, Y. Hiraoka, S. Izumi, M. Kramar, K. Mischaikow and V. Nanda, A topological measurement of protein compressibility, Jpn. J. Ind. Appl. Math., 32 (2015), 1-17.  doi: 10.1007/s13160-014-0153-5.  Google Scholar [13] T. E. Goldberg, Combinatorial Laplacians of simplicial complexes, Senior project, Bard College, 2002. Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.156.3354&rep=rep1&type=pdf. Google Scholar [14] E. Gross, B. Davis, K. L. Ho, D. J. Bates and H. A. Harrington, Numerical algebraic geometry for model selection and its application to the life sciences, J. Roy. Soc. Interface, 13 (2016). doi: 10.1098/rsif.2016.0256.  Google Scholar [15] The GUDHI Project, GUDHI User and Reference Manual, 3.4.1 edition, GUDHI Editorial Board, 2021. Available from: https://gudhi.inria.fr/doc/3.4.1/. Google Scholar [16] W. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. J. Sommese and Y.-T. Zhang, Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1413-1428.  doi: 10.3934/dcdss.2011.4.1413.  Google Scholar [17] W. Hao, B. Hu and A. J. Sommese, Numerical algebraic geometry and differential equations, in Future Vision and Trends on Shapes, Geometry and Algebra, Springer Proc. Math. Stat., 84, Springer, London, 2014, 39–53. doi: 10.1007/978-1-4471-6461-6_3.  Google Scholar [18] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers and P. Virtanen, Array programming with NumPy, Nature, 585 (2020), 357-362.  doi: 10.1038/s41586-020-2649-2.  Google Scholar [19] J. Hauenstein, J. I. Rodriguez and B. Sturmfels, Maximum likelihood for matrices with rank constraints, J. Algebr. Stat., 5 (2014), 18-38.  doi: 10.18409/jas.v5i1.23.  Google Scholar [20] A. Leykin and F. Sottile, Galois groups of Schubert problems via homotopy computation, Math. Comp., 78 (2009), 1749-1765.  doi: 10.1090/S0025-5718-09-02239-X.  Google Scholar [21] L.-H. Lim, Hodge Laplacians on graphs, SIAM Rev., 62 (2020), 685-715.  doi: 10.1137/18M1223101.  Google Scholar [22] X. Liu, X. Wang, J. Wu and K. Xia, Hypergraph-based persistent cohomology (HPC) for molecular representations in drug design, Briefings in Bioinformatics, (2021), bbaa411. doi: 10.1093/bib/bbaa411.  Google Scholar [23] E. R. Love, B. Filippenko, V. Maroulas and G. Carlsson, Topological deep learning, preprint, arXiv: 2101.05778. Google Scholar [24] F. Mémoli, Z. Wan and Y. Wang, Persistent Laplacians: Properties, algorithms and implications, preprint, arXiv: 2012.02808. Google Scholar [25] Z. Meng, D. Vijay Anand, Y. Lu, J. Wu and K. Xia, Weighted persistent homology for biomolecular data analysis, Scientific Reports, 10 (2020), 1-15. doi: 10.1038/s41598-019-55660-3.  Google Scholar [26] F. Nasrin, C. Oballe, D. Boothe and V. Maroulas, Bayesian topological learning for brain state classification, 18th IEEE International Conference On Machine Learning And Applications (ICMLA), Boca Raton, FL, 2019. doi: 10.1109/ICMLA.2019.00205.  Google Scholar [27] D. D. Nguyen, Z. Cang and G.-W. Wei, A review of mathematical representations of biomolecular data, Phys. Chem. Chem. Phys., 22 (2020), 4343-4367.  doi: 10.1039/C9CP06554G.  Google Scholar [28] Y. Ren, J. W. R. Martini and J. Torres, Decoupled molecules with binding polynomials of bidegree $(n, 2)$, J. Math. Biol., 78 (2019), 879-898.  doi: 10.1007/s00285-018-1295-x.  Google Scholar [29] I. Sgouralis, A. Nebenführ and V. Maroulas, A Bayesian topological framework for the identification and reconstruction of subcellular motion, SIAM J. Imaging Sci., 10 (2017), 871-899.  doi: 10.1137/16M1095755.  Google Scholar [30] A. J. Sommese and C. W. Wampler II, The Numerical Solution of Systems of Polynomials. Arising in Engineering and Science, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/5763.  Google Scholar [31] J. Townsend, C. P. Micucci, J. H. Hymel, V. Maroulas and K. D. Vogiatzis, Representation of molecular structures with persistent homology for machine learning applications in chemistry, Nature Communications, 11 (2020), 1-9.  doi: 10.1038/s41467-020-17035-5.  Google Scholar [32] J. Verschelde, Algorithm 795: Phcpack: A general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Softw., 25 (1999), 251-276.  doi: 10.1145/317275.317286.  Google Scholar [33] C. W. Wampler and A. J. Sommese, Numerical algebraic geometry and algebraic kinematics, Acta Numer., 20 (2011), 469-567.  doi: 10.1017/S0962492911000067.  Google Scholar [34] R. Wang, D. D. Nguyen and G.-W. Wei, Persistent spectral graph, Int. J. Numer. Methods Biomed. Eng., 36 (2020), 27pp. doi: 10.1002/cnm.3376.  Google Scholar [35] R. Wang, R. Zhao, E. Ribando-Gros, J. Chen, Y. Tong and G.-W. Wei, HERMES: Persistent spectral graph software, Foundations of Data Science, 3 (2020), 67-97.  doi: 10.3934/fods.2021006.  Google Scholar [36] K. Xia and G.-W. Wei, Persistent homology analysis of protein structure, flexibility, and folding, Int. J. Numer. Methods Biomed. Eng., 30 (2014), 814-844.  doi: 10.1002/cnm.2655.  Google Scholar [37] X.-D. Zhang, The Laplacian eigenvalues of graphs: A survey, preprint, arXiv: 1111.2897. Google Scholar
0-simplex, 1-simplex, 2-simplex and 3-simplex
Rips complexes corresponding to different $r$ values
Pentagon, Heptagon, Octagon and Nonagon
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ against the radius $\alpha$ for the pentagon. The persistent Betti numbers are shown by the blue line. The smallest nonzero eigenvalues calculated by HERMES and Bertini are shown by a red line and red circles, respectively and one can see that they almost coincide. The half edge length is approximately $\sin(\pi/5) \approx 0.58$
(a) A cube with edge length 1. (b) A regular octahedron with edge length $\sqrt{2}$. (c) A regular tetrahedron with edge length $\sqrt{3}$. (d) A regular pyramid with square edge length $\sqrt{2}$ and height 2. (e) A triangular prism with edge length $\sqrt{3}$
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the cube. The length of its face diagonal is $\sqrt{2} \approx 1.4$ and the length of its main diagonal is $\sqrt{3} \approx 1.7$
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the octahedron. Its edge length is $\sqrt{2}$. The circumradius of any face is equal to $\sqrt{2}/\sqrt{3} \approx 0.8$. The circumradius of the octahedron itself is equal to $1$
Some aromatic molecules
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the benzene. The half length of its edge is approximately 0.7Å, and its radius is approximately 1.4Å
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the heptagon
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the octagon
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the nonagon
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the regular tetrahedron with edge length $\sqrt{3}$. The centroid to vertex distance is $3/\sqrt{8} \approx 1.06$
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for a triangular prism. Its base is a regular triangle with edge length $\sqrt{3}$. Its side faces are squares. Important distances are $\sqrt{3}/2, 1, \sqrt{6}/2 \approx 1.22$ and $\sqrt{7}/2 \approx 1.32$
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for a regular pyramid. Its base is a square with edge length $\sqrt{2}$ and its height is 2. Important distances are $\sqrt{2}/2, 1, \sqrt{5}/2 \approx 1.12, 5\sqrt{2}/6 \approx 1.18$ and $5/4$
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the naphthalene
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the anthracene
The illustration of the persistent Betti numbers $\beta_{q}^{\alpha, 0}$ and the smallest nonzero eigenvalues of persistent Laplacians $\lambda_{q}^{\alpha, 0}$ for the pyrene
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